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I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine with the paper up until p149 where MacDonald claims in result 3.9 the following: $$ \omega_{q,t}E_{q,t}\omega_{q,t}^{-1}=E_{t^{-1},q^{-1}} $$

Unfortunately MacDonald provides neither a proof nor a reference for this result. Instead he resorts to borrowing a margin from Fermat's book, claiming to have a proof while not actually giving the proof. Fermat told a better tale, though, claiming to have a "marvelous" proof whereas MacDonald admits to having only a "rather messy" proof.

Does anyone know where I can find MacDonald's "rather messy" proof, or better yet a more elegant proof of this result than the one MacDonald apparently had? Some of the results from this paper seem to also be in MacDonald's earlier book "Symmetric Functions and Hall Polynomials" but I can't find this one in that book either.

I'm trying to gain a rigorous understanding of this paper, but that is difficult for me to do if I need to accept MacDonald's apparent request that I simply gloss over this rather important intermediate step.

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    $\begingroup$ I find the phrase "somewhat seminal" a bit bizarre; either call it seminal or nothing at all. I do not understand the angry tone of the question either. Moreover, I think you could have made a little more effort to spell the name Macdonald correctly. With those comments out of the way -- in Macdonald's paper, equation (3.9) is used to prove the duality theorem 3.1. For another proof of that theorem see Section VI.5 of (at least the Second Edition, 1995) of Macdonald's "Symmetric Functions and Hall Polynomials". Depending on your goal that might be sufficient. $\endgroup$
    – Jules Lamers
    Commented Mar 1, 2021 at 2:52
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    $\begingroup$ If the paper appeared in 1988 but the proof only came later in 1995, then "somewhat seminal" seems a reasonable way to describe it to me. You misunderstood the emotional tone I intended. The intended emotional tone was not anger, but disappointment that Macdonald claimed a result that he didn't provide a proof for. The duality theorem in the paper is 3.5, not 3.1. It does look like the book provides an alternative proof of theorem 3.5, but not necessarily intermediate result 3.9, so I'm exploring that now. Yes, I have the Second Edition of Macdonald's book. Thanks for taking the time to reply! $\endgroup$
    – dash1729
    Commented Mar 1, 2021 at 3:59
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    $\begingroup$ Good. If you're happy with the alternative proof, use it to study what the involution does to $E$: it suffices to understand this on the eigenbasis of Macdonald polynomials. We know that $Q_\lambda(x;q,t) $ differs from $P_\lambda(x;q,t)$ only by a normalizing factor $b_\lambda = 1/\langle P_\lambda,P_\lambda \rangle_{q,t}$, so both are eigenfunctions of the same operator $E$. I would expect that this should be enough to find out what the involution does with the Macdonald operators, and thus with $E = t^{-n} D_n^1 - \sum_i t^{-i}$. $\endgroup$
    – Jules Lamers
    Commented Mar 1, 2021 at 4:15
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    $\begingroup$ I think some of the work by Marshall is focused on this type of questions, arxiv.org/pdf/math/9812080.pdf $\endgroup$
    – Per Alexandersson
    Commented Mar 1, 2021 at 8:20
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    $\begingroup$ Thanks so much for your answers @JulesLamers and PerAlexandersson. Very helpful. Again, sorry for the tone of the initial question--I was just frustrated to get halfway through a paper and find a key intermediate result was lacking a proof! $\endgroup$
    – dash1729
    Commented Mar 2, 2021 at 2:24

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Just to wrap up the discussion in the comments let me note that the answer can be found in Macdonald's Symmetric functions and Hall polynomials (at least in the 2nd edition): a proof of the result of the action of $\omega_{q,t}$ on (a slight modification of) $E_{q,t}$ (which itself is a slight modification of the first Macdonald operator $D_n^1$) is outlined in example (= exercise) 2 of Section VI.3. This is then used in Examples 1 and 2 of Section VI.5 to give an alternative proof of the duality theorem presented in the main text of that section.

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